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### The Grothendieck Ring of Higher Stacks

The Grothendieck ring of varieties is defined to be the free abelian group spanned by isomorphism classes of varieties modulo the cut & paste (or scissor) relations, which say that $[X] = [U] + ...

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### Seeing stacks in the Calculus of Functors

Recently I was told (by an algebraic geometer) that when algebraic geometers look at the Calculus of Functors, they think of stacks.
When I look at the Calculus of Functors, I see a categorification ...

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328 views

### Atiyah-Bott from Beauville-Laszlo

This is a question about the cohomology groups of the stack of vector bundles (with fixed discrete invariants) on an algebraic curve. Explicit formulas for these cohomology groups are known, and they ...

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### Is there a non-quotient stack with affine stabilizers whose good moduli space is a geometric point?

Definitions: One says that a map $\pi\colon\mathcal X\to X$ from an algebraic stack to an algebraic space is a good moduli space if $\pi$ is cohomologically affine and universal for maps to schemes. ...

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551 views

### Stacks in the fpqc topology

This is related to Matt Satriano's earlier question about an analog of Artin's theorem for stacks with an fpqc cover by a scheme.
Suppose one developed the theory of stacks in the fpqc topology and ...

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257 views

### Coarse moduli spaces of stacks for which every atlas is a scheme

Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth projective scheme $P$ by the action of a smooth (finite type separated) reductive group ...

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### Deformations of some simple quotient stacks.

I am interested in stacks of vector bundles on varieties and how deformations of the variety (including non-commutative ones) reflect themselves in deformations of the stack of vector bundles.
I will ...

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### Two questions on algebraic stacks

Question 1: The main reference on algebraic stacks (Laumon and Moret--Bailly) defines a separable algebraic stack as one having universally closed diagonal. For schemes separability is simply defined ...

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### Other examples of the algebro-geometric Ran space

First off, sorry if this seems vague.
Let's recall some definition. Let $X$ be a curve over a field $k$ and $G$ an algebraic group, then the space $Ran_G(X)$ as defined by Lurie in his Tamagawa ...

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283 views

### What is the signficance of the existence of a moduli stack to a moduli problem?

This is a question in pretty unfamiliar territory for me, so if I have conceptual mistakes please correct me.
Let's say we begin with a naive moduli problem: we want a moduli space (whatever space ...

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354 views

### Quasi-coherent sheaves on $M_{FG}$ and the exact functor theorem

I'm struggling with these notes, and one of the things I don't really understand is the following. The notes consider the stack $M_{FG}$ of formal groups; this is the stack associated to the prestack ...

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### Separation condition for higher Deligne-Mumford stacks

Let $X$ be a stack of $n$-groupoids on the site of affine schemes over a fixed base, with the etale topology. If $n=1$ then for $X$ to be Deligne-Mumford, aside from having an etale atlas from an ...

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### Derived Functors, Projection Formula and Base Change in the Derived $\infty$-Category

Let $X$ be a smooth stack and $\mathcal O_X$ the ring of smooth functions on $X$, i.e. for any smooth $M \to X$, $\mathcal O_X(M \to X) = C^\infty(M)$.
In HigherAlgebra, the derived category ...

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127 views

### Is the quotient of a scheme by the free action of an elliptic curve an algebraic space?

Let $X$ be a scheme (I'm happy to assume that $X$ is of finite type, separated, and over $\mathbb{C}$) and let $E$ be an elliptic curve which acts freely on $X$. Does the quotient stack $[X/E]$ have ...

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### Groupoid cardinality of DM stack and point counting on coarse moduli spaces

Let $X$ be a finite type DM stack over a finite field $k$ with a coarse moduli space $X_c$. (We only assume $X_c$ is an algebraic space and $X$ might have infinite inertia stack.)
Under which ...

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### Can the hyperbolic core of a curve over $\mathbb Q$ be defined over $\mathbb Q$ as an algebraic stack

Here is a question I've been wondering about for a while. Currently it is mere curiosity and I do not have any direct applications in mind.
Let $X$ be a smooth quasi-projective geometrically ...

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### KK-witnesses of Gysin maps between differentiable stacks

In 1982 Alain Connes gave the construction of a KK-element $f! \in KK(C(X), C(Y))$ that "witnesses" the fiber integration/Gysin/Umkehr/wrong-way map on topological $K$-theory along a K-orientable map ...

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179 views

### Does every stack with a connection admit an atlas with a connection?

Dear all,
Let $S$ be a scheme in characteristic $0$,
and let $\mathscr{X}/S$ mean a crystal in Artin stacks over $S$ in the sense of this handout, page 4, Definition 0.5, where we replace the scheme ...

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### Maximal algebraic sub-groupoids

By a theorem of Ehresmann, topological and Lie categories (by which I mean categories internal to $Top$ and $Diff$ respectively, with the condition that the source and target in the latter case are ...

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### Nowhere vanishing section of vector bundles over varieties as connectivity of morphism of stacks

The following is, amongst others, a Hartshorne exercise:
Let $V$ be a $k$-variety of dimension $n$ and $\mathcal{E}$ a vector bundle of rank greater than $n$, then, generically, a generating section ...

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### Differential Geometry of (Non-Abelian) Gerbes in the language of Brylinski

Context
In an effort to have a definition for connection on a non-abelian gerbe, in the style of Brylinski, I am reading Breen and Messing's Differential Geometry of Gerbes [BM]. It seems that there ...

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### What is the structure of the stack of complexes supported in dimension less than r?

Let $X$ be something. (smooth and projective variety over C are my assumptions)
The stack $M$ parameterising coherent sheaves on $X$ splits as a disjoint union of open and closed substacks $M_\alpha$, ...

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### Properties of the Zariski-Riemann topology on the set of valuations

One can classify all valuations on a function field $K$ of transcendence degree $2$ over $\mathbf{C}$. Let's consider the set $S_K$ of all valuations on $K$ endowed with the Zariski-Riemann topology.
...

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### Stack cohomology vs cohomology of the quotient

Let $X$ be an affine scheme over an algebraically closed field $K$ of positive characteristic, $G$ a finite group acting on $X$, $[X/G]$ the quotient stack and
$p:[X/G]\to X/G$ the natural map of ...

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### finite etale covering of stacks

If $Y \to X$ is a finite etale map of schemes, then there exists a finite Galois morphism $Z \to X$ (i.e. it's a $Aut(Z/X)$-torsor) that factors as $Z \to Y \to X.$ The case when $X$ is normal is easy ...

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### Properties of finite quotients of quasi-projective varieties

Let $G$ be a finite group acting on a (smooth) quasi-projective variety over $\mathbb C$.
One can consider the stacky quotient $[X/G]$ or the "classical" quotient $X/G$. In general, $[X/G]$ is not a ...

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### Is a stack that is finite etale over an algebraic stack also algebraic?

Here by "algebraic stack" I mean a stack in groupoids over $(\textbf{Sch}/S)_\text{etale}$ (for some scheme $S$) whose diagonal morphism is representable (by schemes), and which is covered by a ...

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### Representable torsors on geometric groupoid

Let $(C,\tau,\mathbb P)$ be a geometric context, as defined by Toen and Vezzosi. Let $(X_1\rightrightarrows X_0)$ be a groupoid object in $C$ such that the source and target morphisms are in $\mathbb ...

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### Cohomology of BG, algebraically

Let $k$ be a field (algebraically closed if you will) and $G$ be a connected reductive group over $k$. I would like to know a purely algebraic computation of the cohomology of $BG$, as the quotient ...

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### Analysis of Eilenberg-MacLane Stacks

In a series of three papers from the fifties, Eilenberg and MacLane did a pretty exhaustive study of what we now call "Eilenberg-MacLane spaces" and used a lot of machinery to do it, e.g. Whitehead's ...

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### Where does the ''hyper'' go?

There is a model structure on the category $Grpd$ of groupoids such that weak equivalences are equivalences of categories, cofibrations are the injections on objects (see nlab) and there is a Quillen ...

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### Chow ring of a $\mu_2$-gerbe

Suppose that $X$ is a stack, and $Y \to X$ is a $\mu_2$-gerbe. Is there any relationship between the integral Chow rings (in the sense of Edidin and Graham) of $X$ and $Y$?
(I assume they become ...

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### How does the machinery of left-exact comonads generalize from sheaves to stacks?

Suppose that we have two Grothendieck sites, their associated sheaves $\mathcal{E}=\rm{Sh}(\bf{C},J)$ and $\mathcal{F}=\rm{Sh}(\bf{D},K)$ and a geometric surjection $f:\mathcal{E}\to\mathcal{F}$. This ...

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### Definition of derived category of a stack

In their book, Bernstein an Lunts define the equivariant derived category in several ways. One can be expressed as follows:
Let $X$ be a say complex variety with an action by an algebraic group $G$.
...

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### Cohomology of line bundles on smooth projective toric Deligne-Mumford stacks

Let $\mathcal{X}$ be a smooth projective toric Deligne-Mumford stack with coarse moduli scheme $\pi\colon \mathcal{X}\rightarrow X.$
Let $[D]$ be a Cartier divisor on $\mathcal{X}$ such that ...

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### properness of stack

Hi,
assume we have an algebraic stack $A$ over $Sch(\mathbb{Z})$ which is quasi-compact and with separated diagonal. Assume that I have a stack $B$ which is obtained by rigidifying $A$ by a subgroup ...

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### Derived pullback of quasi-coherent complexes between algebraic stacks

Let $X$ and $Y$ be Artin algebraic stacks, and $f:X\to Y$ a morphism. I am interested in a pullback morphism
$Lf^\ast : D^-_{qcoh}(Y)\to D^-_{qcoh}(X)$.
In his paper Sheaves on Artin stacks, ...

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### When is a category of groupoid schemes fibred over schemes?

The category of topological categories $Cat(Top)$ is fibred over $Top$ - the functor sending a groupoid $X_1 \rightrightarrows X_0$ to its object space $X_0$ is a Grothendieck fibration. Now one can ...

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### When is the base change morphism an isomorphism?

This is a rewrite of a previous question, which was in turn a follow up question to Leray-Hirsch principle for étale cohomology The motivation is to clarify some points in Torsten Ekedahl's ...

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### quotient a scheme by a stratified vector bundle

Let $k$ be a field.
Let $X$ be a $k$-scheme of finite type, normal and integral. We consider $f,g:R\rightarrow X$ an equivalence relation, surjective and such that it is a stratified vector bundle, ...

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### Is the upper half plane an algebraic stack?

Here by algebraic stack I mean an algebraic stack over the etale site $\textbf{Sch}/\mathbb{C}$.
So I've read from various nonrigorous sources that the upper half plane $\mathcal{H}$ is a fine moduli ...

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### Is the cotangent complexes of groupoids bounded above by degree $1$?

Let $\mathcal{X}$ be a stack given by a groupoid $X_1\rightrightarrows X_0$, where $X_0$ and $X_1$ are smooth $k$-varieties. Let $\mathbb{L}_{\mathcal{X}/k}$ be the cotangent complex of ...

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### Properties of the induced map between inertia stacks

Let $\mathcal X$ and $\mathcal Y$ be (separated) Deligne-Mumford stacks. A morphism of stacks $f:\mathcal X \to \mathcal Y$ induces a morphism between inertia stacks $\tilde f:I\mathcal X \to ...

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### Quasi-finite morphisms of stacks

Let $f:X\to Y$ be a morphism of ``nice" stacks over $\mathbf C$ such that the induced morphism on coarse moduli spaces is quasi-finite. Is $f$ quasi-finite?
By a "nice" stack I mean a smooth finite ...

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### what's the coherent sheaf and sheaf cohomology on Deligne-Mumford stack ?

hello, everyone, I am reading Deligne-Mumford's paper, and the section 4 of this paper, they given some properties of Deligne-Mumford stack without proof. I can't understand the coherent sheaf and ...

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### Is the category of quasi-coherent sheaves on a concentrated stack locally finitely presentable?

Let's call an Artin stack $X$ concentrated iff it is quasi-compact and quasi-separated (the latter usually being included in the definition of an Artin stack). The category of quasi-coherent sheaves ...

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### Coarse Moduli space of proper Deligne-Mumford stacks

Let F be a finite type proper Deligne-Mumford Stack over a perfect field. Is it true that the coarse moduli space of F is proper?

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### Why $D^b(S)\cong D^b_{\text{Car}}(\text{cosq}(X\rightarrow S))$?

Let $p: X\rightarrow S$ be a map between topological spaces and we can construct the simplicial space $\text{cosq}(X\rightarrow S)$ where $X_0=X$, $X_1=X\times_S X$ and
$$
X_n=\underbrace{ X\times_S ...

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### Zariski's Main Theorem for stacks

Let $X$ be a separated Deligne Mumford stack of finite type over a base field $k$ and $Y$ be a proper Deligne Mumford stack over $k$.
Assume there is a quasifinite, representable, surjective and ...

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### Irreducibility of monodromy of eigenspaces of families of cyclic coverings

In the article "La conjecture de Weil", Deligne proves that for the primitive cohomology of a universal family $f:X \rightarrow S$ for $M_{d,n}$ the moduli stack of hypersurfaces of degree $d$ in ...